Reading between the rational sections: Global structures of 4d N=2 KK theories

We study how the global structure of rank-one 4d N=2 𝒩=2 supersymmetric field theories is encoded into global aspects of the Seiberg-Witten elliptic fibration. Starting with the prototypical example of the su(2) 𝔰𝔲(2) gauge theory, we distinguish between relative and absolute Seiberg-Witten curves. For instance, we discuss in detail the three distinct absolute curves for the SU(2) SU(2) and SO(3)_± SO(3)± 4d N=2 𝒩=2 gauge theories. We propose that the 1 1 -form symmetry of an absolute theory is isomorphic to a torsion subgroup of the Mordell-Weil group of sections of the absolute curve, while the full defect group of the theory is encoded in the torsion sections of a so-called relative curve. We explicitly show that the relative and absolute curves are related by isogenies (that is, homomorphisms of elliptic curves) generated by torsion sections - hence, gauging a one-form symmetry corresponds to composing isogenies between Seiberg-Witten curves. We apply this approach to Kaluza-Klein (KK) 4d N=2 𝒩=2 theories that arise from toroidal compactifications of 5d and 6d SCFTs to four dimensions, uncovering an intricate pattern of 4d global structures obtained by gauging discrete 0 0 -form and/or 1 1 -form symmetries. Incidentally, we propose a 6d BPS quiver for the 6d M-string theory on R^4× T^2 ℝ4×T2 .